1. Field of Invention
The present invention relates to the degree of nonlinearity. More particularly, the present invention relates to quantifying the degree of nonlinearity.
2. Description of Related Art
The term ‘nonlinearity’ has been loosely used, in most cases, not to clarify but as a fig leaf to hide our ignorance. As a result, any anomaly without obvious and ready explanations is labeled as “nonlinear effect”. Such an approach could certainly be right for some cases, but, at the best, the answer is is not complete, for the nonlinear effects could have many causes: For example, nonlinear effects could be the results of the intrinsic properties of the system and also arise from the various initial conditions. Unfortunately, under the present state of our understanding, no better solution is available. The central of the problem is that the definition of ‘nonlinearity’ is only qualitative. Let us review the definition of nonlinearity.
Currently the nonlinearity is defined based a system point of view. And the definition is not directly on nonlinearity, but on linearity; any system that is not linear would be nonlinear. The linearity is based on linear algebra: For any system L is linear, if for inputs x1 and x2 we have outputs y1 and y2 respectively as:L(x1)=y1;L(x2)=y2;  (1)
We should also haveL(αx1+βx2)=αy1+βy2  (2)
Any system does not satisfy this superposition and scaling rules given in Equation (1) and (2) is a nonlinear system. This definition is rigorous but qualitative, for the answer offer is ‘yes’ or ‘no’ without any quantitative measure. Furthermore, this system approach may not be very practical. For many phenomena, it might be difficult, if not impossible, to define the system in close form to test this input vs. out put approach. For example, many natural systems are so complicated that a closed system is hard to define. Even for those systems that we could find explicit analytic expression in closed form, the inputs and out puts could still be hard to define. For example, for the autonomous system, the state of the phenomena depends totally on the initial condition, which is hardly qualified as inputs. The motion or the flow of the system is also hard to be treated as output. Furthermore, for complicate system, there might not be a small parameter for us to track the progress of the system from linear to nonlinear as that parameter changes. All of these difficulties made the discussions of nonlinearity hard to quantify. Without quantification, however, it would be impossible to discuss the nonlinearity more precise and prevent its loose usage. A brief review of the state-of-the-arts nonlinearity tests is given first.
A Brief Review of the State-of-the-arts in Nonlinearity Tests
Even though the definition of nonlinearity used now is qualitative, to be able to determine whether a given set of data is or is not coming from a nonlinear process s still a highly desirable. Many tests have been proposed and used, most frequently in financial data (see, for example, Kantz and Schreiber, 1997 and Tsay, 2005); they all have some deficiencies. The current available nonlinearity teat is briefly summarized here:
a. Tests Based on Inputs and Outputs
As discussed above, the inputs and outputs are hard to define for most cases. There are, however, special occasions when this is possible in a closed system as in some engineering applications. Methods to identify the nonlinearity could be based on the inputs and outputs data. In principle, this is a much easier problem. This approach is best summarized by Bendat (1990). The main tools employed were the probability distribution functions and higher order Fourier spectral analysis. In theory, the approach would be possible, for the probability distribution changes from inputs to outputs could be well defined through the generalized theorem of probability, which states that known the distribution of a random variable, one could derive the distribution of any function of that variable. Either probability or the spectral approach all depends on the stationary assumption. Furthermore, when the processes are nonlinear, the Fourier spectral analysis could also generate spurious harmonics. Finally, the best result this approach could provide is still limited to qualitative ones. When the closed system is not available as is the rule rather than exception, this approach is totally infeasible. Therefore, we have to study the cases when only outputs data is available.
b. Tests Totally Based on Data.
One of the oldest nonlinearity test is the purely data based method of higher order spectral analysis (Nikias and Petropulu, 1993). The idea is to use cumulant spectra, such as the bispectra and the trispectra. There are two major problems for this approach: First, the cumulant spectra do not measure nonlinearity directly; they actually measure the departure from normalcy. For example, the total integration of a bispectrum gives the skewness and the integration of tri-spectrum, the kurtosis. Granted that linear processes are mostly Gaussian by virtue of the central limit theorem, but nonlinear process could also produce data with Gaussian distribution. The obvious case is the homogeneous turbulence, a phenomenon of the fluid motion produced through the instability of nonlinear flows and highly interactive cascade of eddies (Batchelor, 1953; Frisch, 1995), yet the distribution of its velocity is normal. Secondly, the computation employs the Fourier transform, which is under the assumption of stationarity. This will strongly restrict the applicability of the test and weaken its test results.
c. Nonparametric Tests
In this approach, all the tests are based on the hypothesis that the data from linear processes should have near linear residue from a properly defined linear model. Famous among the tests are the ones proposed by McLeod and Li (1983), where ARMA (p,q) model is used; Brock, Dechert and Scheikman (BDS, 1987), where ‘correlation integral’ is used to test the iid (independent, identical distribution) of the data distribution; and Theiler et al (1992), where an ensemble of linear surrogate sets of data is set up based on a null hypothesis then compare with the given data. Here, the BDS test is not designed for nonlinearity at all; it is for testing of whether the data is stochastic or chaotic. Theiler's tests depend on the null hypothesis and the ensemble of models so established. The ensemble could be extensive, but hardly exhaustive. All the tests are again qualitative: simply determining the process is or is not linear.
d. Parametric Tests
The parametric approach is based on a given model, such as auto-regression model. Ramsey (1969) first proposed a test for linear least square regression analysis. Although later generalization by Keenan (1985) and Tsay (1986) using different regressors, the test is limited to specific model.
For more detailed discussion, one should consult the original paper, or see Kantz and Schreiber (1997) and Tsay (2005). Critical shortcomings of all the tests are that most of the tests are limited to stationary processes, and also that all the answers are still qualitative in nature. Few problems could be answered satisfactorily with a qualitative answer. If nonlinearity is an issue, quantitative answer is imperative.